integral of (1/(cos^2(x))-1)os(x)

解答

\int (\frac{1}{cos ^{2} ( x )} - 1) d cos (x) d x

d (ln ∣ tan (x) + sec (x) ∣ - sin (x)) + C

求解步骤

\int (\frac{1}{cos ^{2} ( x )} - 1) d cos (x) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= d \cdot \int (\frac{1}{cos ^{2} ( x )} - 1) cos (x) d x

化简

(\frac{1}{cos ^{2} ( x )} - 1) cos (x) : sin (x) tan (x)

= d \cdot \int sin (x) tan (x) d x

使用三角恒等式改写

= d \cdot \int \frac{1 - cos ^{2} ( x )}{cos ( x )} d x

乘开

\frac{1 - cos ^{2} ( x )}{cos ( x )} : \frac{1}{cos ( x )} - cos (x)

= d \cdot \int \frac{1}{cos ( x )} - cos (x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= d (\int \frac{1}{cos ( x )} d x - \int cos (x) d x)

\int \frac{1}{cos ( x )} d x = ln ∣ tan (x) + sec (x) ∣

\int cos (x) d x = sin (x)

= d (ln ∣ tan (x) + sec (x) ∣ - sin (x))

解答补常数

= d (ln ∣ tan (x) + sec (x) ∣ - sin (x)) + C

\int \frac{( x ^{2} + 3 )}{x} d x

\int 3 3 x^{2} d x

\int \frac{1}{x ^{2} + 6 x + 8} d x

\int \frac{x ^{3} - 3 x}{x ^{4} - 6 x ^{2} - 9} d x

\int - \frac{2 x}{( x ^{2} + 1 ) ( x + 1 )} d x